The Barles–Souganidis (1991) convergence theorem states that a numerical scheme for a fully nonlinear second-order PDE converges uniformly to the unique viscosity solution provided the following conditions hold:
- Monotonicity
- Stability
- Consistency
- PDE that satisfies a comparison principle
While most conditions are automatically satisfied in many economic models, economists must be very careful with the monotonicity condition, which may fail in some cases and hard to detect at first glance.
Notations#
Let’s consider the following PDE of \(v(x):\mathbb{R}^N\to\mathbb{R}\). For simplicity, I shut down all uncertainties (it is trivial to realize that adding diffusion term won’t change the conclusion).
$$ \begin{aligned} & \rho v(x) = u(x) + \sum_{i=1}^N \mu_i(x) \cdot \frac{\partial v}{\partial x_i}(x) \end{aligned} $$Let’s consider a grid \(\mathcal{X}\) over a subset of \(\mathbb{R}^N\). For a point \(x\in \mathbb\)
Solution#
Reference#
Barles and P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic analysis, 4(3):271–. 283, 1991.